Accompanying dataset for the paper "Multi-material topology optimization of structural load-bearing capacity using limit analysis"

This dataset contains simulation results supporting the following paper:

Mourad, L., Bleyer, J., Mesnil, R., Nseir, J., Sab, K., & Raphael, W. (2025). Multi-material topology optimization of structural load-bearing capacity using limit analysis. Journal of Theoretical, Computational and Applied Mechanics - https://doi.org/10.46298/jtcam.12427

Simulation code

The code used to generate the simulation results can be found at

demos/topology_optimization · master · navier-fenics / fenics-optim · GitLab

Dataset description

Scripts

Jupyter notebook and raw data associated with Figures 9 and 15 are available in the scripts.

MBB example

Reinforcement optmization of the MBB example for various reinforcement strength criteria:

Fig.5a: MBB_beam/reinforcement_vonMises_fc_1.0_ft_1.0

Fig.5b: MBB_beam/reinforcement_fc_1.0_ft_1.0

Fig.5c: MBB_beam/reinforcement_no_compression_fc_1.0_ft_1.0

Fig.5d: MBB_beam/reinforcement_orthogonal_fc_1.0_ft_1.0

Bi-material load-maximization of the MBB example

Fig.6a: MBB_beam/rankine_vonMises_frac_0.2

Fig.6b: MBB_beam/rankine_L1Rankine_frac_0.2

Bi-material load-maximization of the MBB example with tension/compression splitting for various volume fractions

Fig.7a: MBB_beam/l1rankine_l1rankine_frac_0.05

Fig.7b: MBB_beam/l1rankine_l1rankine_frac_0.1

Fig.7c: MBB_beam/l1rankine_l1rankine_frac_0.2

Fig.7d: MBB_beam/l1rankine_l1rankine_frac_0.3

Bi-material load-maximization of the MBB example with discrete orientations for the tensile phase

Fig.8a: MBB_beam/fc_1.0_ft_1.0_0_deg

Fig.8b: MBB_beam/fc_1.0_ft_1.0_30_deg

Fig.8c: MBB_beam/fc_1.0_ft_1.0_45_deg

Fig.8d: MBB_beam/fc_1.0_ft_1.0_90_deg

Bridge example

Symmetric strengths fc=ft=1

Fig.11a: bridge/frac_02_bimaterial_fc_1.0_ft_1.0

Fig.11b: bridge/frac_02_single_material_fc_1.0_ft_1.0

Asymmetric strengths fc=1, ft=10

Fig.12a: bridge/frac_02_bimaterial_fc_1.0_ft_10.0

Fig.12b: bridge/frac_02_single_material_fc_1.0_ft_10.0

Asymmetric strengths fc=1, ft=10

Fig.13a: bridge/frac_02_bimaterial_fc_10.0_ft_1.0

Fig.13b: bridge/frac_02_single_material_fc_10.0_ft_1.0

Bridge example with varying density-cost $\omega$ in the case fc=ft=1

Fig.14a: bridge/bimaterial_fc_1.0_ft_1.0_cost_0.1

Fig.14b: bridge/bimaterial_fc_1.0_ft_1.0_cost_0.25

Fig.14c: bridge/bimaterial_fc_1.0_ft_1.0_cost_0.5

Fig.14d: bridge/bimaterial_fc_1.0_ft_1.0_cost_0.75

Fig.14e: bridge/bimaterial_fc_1.0_ft_1.0_cost_0.9

Deep beam example

Deep beam example with inclined reinforcements (bi-material)

Fig. 17a: deep_beam/muttoni_bivolmin_iso

Deep beam example with orthogonal reinforcements (bi-material)

Fig. 18a: deep_beam/muttoni_bivolmin_ortho

einforcement optimization of the deep beam exampl

Fig. 19a: deep_beam/muttoni_reinfvolmin_iso

Fig. 19b: deep_beam/muttoni_reinfvolmin_ortho

JTCAM Paper Abstract

We extend the problem of finding an optimal structure with maximum load-bearing capacity to the case of multiple materials. We first consider a reinforcement optimization case where the structure consists of a fixed background matrix material with given strength properties and optimize the reinforcement topology within this material. We discuss the use of various isotropic and anisotropic strength criteria to model the reinforcing phase, including reinforcements with discrete orientations. In a second time, we investigate a bi-material formulation where we optimize the topology of two material phases simultaneously. Various choices for the material strength conditions are proposed and we apply this formulation to the optimization of pure tensile and compressive phases of a single material. In all cases, two optimization variants are proposed using concepts of convex optimization and limit analysis theory, namely maximizing the load-bearing capacity under a fixed volume constraint or minimizing the volume under a fixed loading. Both problems are convex and a penalization procedure is proposed. The underlying problems can be solved using conic programming solvers. Illustrative applications demonstrate the versatility of the proposed formulation, including the influence of the selected strength criteria, the possibility to obtain structures with members of fixed orientation or structures with different importance granted to tensile and compressive regions. Finally, we also draw a parallel with the generation of strut-and-tie models for the analysis of reinforced concrete structures.